(Solved): Question 2 [25 points] a) The distribution (CDF) of a discrete random variable is shown in the fig ...

Question 2 [25 points] a) The distribution (CDF) of a discrete random variable is shown in the figure below. i) Find the probability \( \operatorname{Pr}(2 \leq X \leq 4) \). ii) Calculate the expected value of \( X \), ie \( \mathrm{E}[X] \). iii) Sketch the corresponding probability mass function PMF. b) The PDF of a random variable \( \mathrm{X} \) is given by: \[ f_{X}(x)=\left\{\begin{array}{cc} 0.4+k x, & 0 \leq x \leq 4 \\ 0 & \text { otherwise } \end{array}\right. \] i) Find the value \( k \) that makes \( f_{x} \) a valid PDF. ii) Find the CDF, \( F_{x}(\mathbf{x}) \). Question 1 [25 points] (a) Answer the following questions (True or False). (i) If \( A \) and \( B \) are both nonempty events of a sample space \( S \) and \( A \) and \( B \) are mutually exclusive, then \( A \) and \( B \) are dependent. (ii) The CDF of a random variable \( X \) is bounded by \( -1 \leq F_{X}(x) \leq 1 \). (iii) If \( X \) is bell-shaped and symmetric, then Chebyshev's rule does not apply. (iv) According to the central limit theorem, the distribution function of the sum of a large number of random variables approached a Gaussian distribution. (v) The power spectral density of a WSS random process is defined as The Fourier transform of the auto-correlation function of the random process. (b) A student decided to make random guesses for all the above 5 true/false questions. What is the probability of this student getting at least 2 correct answers? (c) A missile can be accidentally launched if two relays A and B both have failed. The probabilities of A and \( B \) failing are known to be \( 0.09 \) and \( 0.03 \) respectively. It is also known that \( B \) is more likely to fail (probability \( 0.05 \) ) if A has failed. (i) What is the probability of an accidental missile launch? (ii) Are the events "A fails" and "B fails" statistically independent?